A class of residual distribution schemes and their relation to relaxation systems

JOURNAL OF COMPUTATIONAL PHYSICS

卷:227

Article

Rossmanith, James A.

关键词: Residual distribution schemes;    Relaxation systems;    Approximate Riemann solvers;    Finite volume methods;    Hyperbolic conservation laws;   

DOI  :  10.1016/j.jcp.2008.07.007

来源: Elsevier

【 摘 要 】

Residual distributions (RD) schemes are a class of high-resolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal I D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [R.J. LeVeque, M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes, J. Comput. Phys. 172 (2001) 572] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, Lax-Friedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [S. Jin. Z.P. Xin, Relaxation schemes for systems of conservation-laws in arbitrary space dimensions, Commun. Pure Appl, Math. 48 (1995) 235]. In this work we extend LeVeque and Pelanti's results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard N-scheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local Lax-Friedrichs scheme. This new Riemann solver does not require the use Roe-Struijs-Deconinck averages, nor does it require the inversion of an m x m matrix in each computational grid cell, where m is the number of conserved variables. Once this new scheme is established, we apply it on a few standard cases for the 2D compressible Euler equations of gas dynamics. We show that through the use of linear-preserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the N-scheme, despite being computationally less expensive. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free